1. An intersection problem of graph-directed attractors and an application
- Author
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Ruan, Huo-Jun and Xiao, Jian-Ci
- Subjects
Mathematics::Dynamical Systems ,Primary 28A80, Secondary 54A05 ,General Topology (math.GN) ,FOS: Mathematics ,Mathematics::General Topology ,Mathematics - General Topology - Abstract
Let $(K_1,\ldots,K_n)$ be a Cantor type graph-directed attractors in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion of whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of the attractor's geometric approximations. The method is also applicable for more general graph-directed systems. As an application, we are able to determine the connectedness of all $d$-dimensional generalized Sierpi\'nski sponges of which the corresponding IFSs are allowed to have rotational and reflectional components., Comment: 24 pages, 4 figures
- Published
- 2022
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